Although equations of the type (REF ) are generally
not gradient flows with respect to the standard Wasserstein metric when \(v \ne 1\) , they can be interpreted as gradient-flow evolutions of \(\mathcal {F}\)  with respect to a generalised Wasserstein distance which takes into account the non-linearity. Such point of view was introduced in [1]} and [2]} and further investigated in [3]}, while some applications of the corresponding gradient-flow structure may be found in the study
of non-linear cross-diffusion systems [4]} and fourth-order equations, as Cahn-Hillard and thin films equations (see [5]}, [6]}).
However, being the generalised Wasserstein metric only expressed in a dynamical implicit formulation, it is not easy to provide the corresponding generalization of the Jordan-Kinderlehrer-Otto scheme and construct solutions of (REF ) in this spirit. The validation of a scalar transport equation with nonlinear mobility, which can be also regarded as a scalar nonlinear conservation law, can then be obtained proving the convergence of systems of fully deterministic interacting particles. A first rigorous result dealing with pure linear drift can be found in [7]}, while the case of external potentials with symmetries has been considered in [8]}.
A deterministic many-particle limit result for scalar evolutions with non-local interactions was initially performed in [9]} in the smooth and pure aggregative regime. In [10]} and [11]} the case of mildly singular non-local interaction kernels (as Morse and Yukawa type ones) is considered.
The authors of [10]} work directly at the level of the gradient-flow structure at the continuous time-scale exploiting an energy-dissipation balance argument, while more general time-dependent kernels are included in the analysis in [11]} thanks to the introduction of a new integrated version of the deterministic particle scheme.
On the other hand, the first deterministic many-particle limit for scalar non-local transport equations as in (REF ) with nonlinear mobility and a diffusive term was considered in [14]}. The compactness argument employed in [14]} resembles the techniques used in [9]} and it is based on strong \(BV\)  estimates and a good control on the support of the moving particles. Since the presence of the diffusive term in (REF ) plays against the latter bound, the analysis in [14]} is carried out inside bounded intervals with zero velocity boundary conditions and for initial data enjoying the no vacuum condition.
